POP247 - campus EDA
Models for St Lucia - Wi-Fi & Green Bridge (pedestrians + cyclists)
24 June, 2022
1 Pedestrian & bike counts
1.1 Original data
Detailed example counts for week of 2018-12-03 across
type and direction.
Overview of months, cyclists, outbound
1.2 Stocks on campus
1.2.1 Without modification
1.2.2 With midnight ‘reset’
Frequency of net from whole period
Frequency of stocks from whole period
2 Wi-Fi
2.1 Original data
Detailed example counts for week of 2018-12-09 across
type of data.
Overview of months, connected users
2.2 Distribution of counts
Frequency of counts from whole period
2.3 Distribution fits
3 Time series comparisons
3.1 Cyclists
3.1.1 Inbound, connected users
3.1.2 Outbound, connected users
3.2 Pedestrians
3.2.1 Inbound, connected users
3.2.2 Outbound, connected users
3.3 Stocks, connected users
4 Correlations, connected users
5 Models
5.1 INLA setup
# ####
# Jeffreys prior
a1 <- 5e-5
b1 <- 5e-5
lgprior1 <- list(prec = list(param = c(a1, b1)))
# Gelman prior
a2 <- -0.5
b2 <- 5e-5
lgprior2 <- list(prec = list(param = c(a2, b2)))
# iid prior
# Schrödle & Held 2010 & Blangiardo et al 2013
a0 <- 1
b0 <- 0.1
prior.nu <- list(prec = list(param = c(a0, b0)))
# intercept & fixed
inla.set.control.fixed.default()
# intercept ~ N(0,0)
# other fixed effects ~ N(0, 0.001)
#
# where the format is N(mean, precision)
# precision = inverse of the variance.
# PC prior
U <- 1
hyper.prec = list(theta = list(
prior = "pc.prec",
param = c(U, 0.01)
))
# scaling
inla.setOption(scale.model.default = TRUE)5.2 RW1
5.2.1 One trend for stocks
mod1 <- inla(users_mac ~
f(stock, model = "rw1", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.663, Running = 5.36, Post = 0.24, Total = 6.26
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.823 0.008 7.806 7.823 7.839 7.822 0
Random effects:
Name Model
stock RW1 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 5.21 0.100 5.02
Precision for stock 2.33 0.353 1.73
0.5quant 0.975quant mode
size for the nbinomial observations (1/overdispersion) 5.21 5.41 5.21
Precision for stock 2.30 3.11 2.24
Deviance Information Criterion (DIC) ...............: 81065.63
Deviance Information Criterion (DIC, saturated) ....: 155235.61
Effective number of parameters .....................: 119.67
Watanabe-Akaike information criterion (WAIC) ...: 81078.70
Effective number of parameters .................: 127.84
Marginal log-Likelihood: -42295.32
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.2.2 One trend for stocks + weekday
mod2 <- inla(users_mac ~
weekday +
f(stock, model = "rw1", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.533, Running = 5.61, Post = 0.237, Total = 6.38
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.851 0.017 7.818 7.851 7.884 7.851 0
weekdayMonday -0.061 0.023 -0.106 -0.061 -0.015 -0.061 0
weekdaySaturday -0.073 0.025 -0.122 -0.073 -0.025 -0.073 0
weekdaySunday -0.076 0.025 -0.124 -0.076 -0.027 -0.076 0
weekdayThursday 0.025 0.023 -0.020 0.025 0.071 0.025 0
weekdayTuesday -0.019 0.023 -0.065 -0.019 0.027 -0.019 0
weekdayWednesday -0.053 0.023 -0.098 -0.053 -0.008 -0.053 0
Random effects:
Name Model
stock RW1 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 5.23 0.100 5.04
Precision for stock 2.36 0.357 1.75
0.5quant 0.975quant mode
size for the nbinomial observations (1/overdispersion) 5.23 5.43 5.23
Precision for stock 2.33 3.14 2.26
Deviance Information Criterion (DIC) ...............: 81047.47
Deviance Information Criterion (DIC, saturated) ....: 155217.45
Effective number of parameters .....................: 125.26
Watanabe-Akaike information criterion (WAIC) ...: 81064.40
Effective number of parameters .................: 136.69
Marginal log-Likelihood: -42324.36
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.2.3 One trend for stocks + weekend + weekday
mod3 <- inla(users_mac ~
weekend +
weekday +
f(stock, model = "rw1", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.515, Running = 6.61, Post = 0.251, Total = 7.37
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.851 0.017 7.818 7.851 7.884 7.851 0
weekendWeekend -0.049 18.258 -35.896 -0.050 35.768 -0.049 0
weekdayMonday -0.061 0.023 -0.106 -0.061 -0.015 -0.061 0
weekdaySaturday -0.024 18.258 -35.871 -0.025 35.793 -0.024 0
weekdaySunday -0.026 18.258 -35.873 -0.027 35.790 -0.026 0
weekdayThursday 0.025 0.023 -0.020 0.025 0.071 0.025 0
weekdayTuesday -0.019 0.023 -0.065 -0.019 0.027 -0.019 0
weekdayWednesday -0.053 0.023 -0.098 -0.053 -0.008 -0.053 0
Random effects:
Name Model
stock RW1 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 5.23 0.100 5.04
Precision for stock 2.36 0.357 1.75
0.5quant 0.975quant mode
size for the nbinomial observations (1/overdispersion) 5.23 5.43 5.23
Precision for stock 2.32 3.15 2.26
Deviance Information Criterion (DIC) ...............: 81047.43
Deviance Information Criterion (DIC, saturated) ....: 155217.40
Effective number of parameters .....................: 125.20
Watanabe-Akaike information criterion (WAIC) ...: 81064.40
Effective number of parameters .................: 136.68
Marginal log-Likelihood: -42324.90
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.3 RW2
5.3.1 One trend for stocks
mod5 <- inla(users_mac ~
# weekend +
# weekday +
f(stock, model = "rw2", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.521, Running = 7.01, Post = 0.175, Total = 7.7
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.823 0.008 7.806 7.823 7.839 7.823 0
Random effects:
Name Model
stock RW2 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 5.114 0.097 4.923
Precision for stock 0.019 0.003 0.013
0.5quant 0.975quant
size for the nbinomial observations (1/overdispersion) 5.113 5.307
Precision for stock 0.019 0.027
mode
size for the nbinomial observations (1/overdispersion) 5.114
Precision for stock 0.018
Deviance Information Criterion (DIC) ...............: 81111.96
Deviance Information Criterion (DIC, saturated) ....: 155281.94
Effective number of parameters .....................: 57.13
Watanabe-Akaike information criterion (WAIC) ...: 81111.69
Effective number of parameters .................: 55.84
Marginal log-Likelihood: -45505.61
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.3.2 One trend for stocks + weekday
mod6 <- inla(users_mac ~
# weekend +
weekday +
f(stock, model = "rw2", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.563, Running = 6.52, Post = 0.234, Total = 7.31
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.854 0.017 7.821 7.854 7.888 7.854 0
weekdayMonday -0.074 0.023 -0.120 -0.074 -0.029 -0.074 0
weekdaySaturday -0.074 0.025 -0.123 -0.074 -0.026 -0.074 0
weekdaySunday -0.079 0.025 -0.128 -0.079 -0.031 -0.079 0
weekdayThursday 0.024 0.023 -0.022 0.024 0.069 0.024 0
weekdayTuesday -0.020 0.023 -0.065 -0.020 0.026 -0.020 0
weekdayWednesday -0.056 0.023 -0.102 -0.056 -0.011 -0.056 0
Random effects:
Name Model
stock RW2 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 5.14 0.098 4.945
Precision for stock 0.02 0.003 0.014
0.5quant 0.975quant
size for the nbinomial observations (1/overdispersion) 5.139 5.332
Precision for stock 0.019 0.027
mode
size for the nbinomial observations (1/overdispersion) 5.141
Precision for stock 0.019
Deviance Information Criterion (DIC) ...............: 81090.97
Deviance Information Criterion (DIC, saturated) ....: 155260.95
Effective number of parameters .....................: 63.03
Watanabe-Akaike information criterion (WAIC) ...: 81094.76
Effective number of parameters .................: 65.46
Marginal log-Likelihood: -45532.49
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.3.3 One trend for stocks + weekend + weekday
mod7 <- inla(users_mac ~
weekend +
weekday +
f(stock, model = "rw2", scale.model = TRUE, hyper = hyper.prec),
family = "nbinomial",
control.compute = list(dic= TRUE, waic = TRUE),
data = data)
Call:
c("inla.core(formula = formula, family = family, contrasts = contrasts,
", " data = data, quantiles = quantiles, E = E, offset = offset, ", "
scale = scale, weights = weights, Ntrials = Ntrials, strata = strata,
", " lp.scale = lp.scale, link.covariates = link.covariates, verbose =
verbose, ", " lincomb = lincomb, selection = selection, control.compute
= control.compute, ", " control.predictor = control.predictor,
control.family = control.family, ", " control.inla = control.inla,
control.fixed = control.fixed, ", " control.mode = control.mode,
control.expert = control.expert, ", " control.hazard = control.hazard,
control.lincomb = control.lincomb, ", " control.update =
control.update, control.lp.scale = control.lp.scale, ", "
control.pardiso = control.pardiso, only.hyperparam = only.hyperparam,
", " inla.call = inla.call, inla.arg = inla.arg, num.threads =
num.threads, ", " blas.num.threads = blas.num.threads, keep = keep,
working.directory = working.directory, ", " silent = silent, inla.mode
= inla.mode, safe = FALSE, debug = debug, ", " .parent.frame =
.parent.frame)")
Time used:
Pre = 0.538, Running = 6.88, Post = 0.24, Total = 7.66
Fixed effects:
mean sd 0.025quant 0.5quant 0.975quant mode kld
(Intercept) 7.854 0.017 7.821 7.854 7.888 7.854 0
weekendWeekend -0.052 18.257 -35.896 -0.052 35.763 -0.052 0
weekdayMonday -0.074 0.023 -0.120 -0.074 -0.029 -0.074 0
weekdaySaturday -0.023 18.257 -35.867 -0.023 35.792 -0.023 0
weekdaySunday -0.028 18.257 -35.872 -0.028 35.787 -0.028 0
weekdayThursday 0.024 0.023 -0.022 0.024 0.069 0.024 0
weekdayTuesday -0.020 0.023 -0.065 -0.020 0.026 -0.020 0
weekdayWednesday -0.056 0.023 -0.102 -0.056 -0.011 -0.056 0
Random effects:
Name Model
stock RW2 model
Model hyperparameters:
mean sd 0.025quant
size for the nbinomial observations (1/overdispersion) 5.14 0.098 4.946
Precision for stock 0.02 0.003 0.014
0.5quant 0.975quant
size for the nbinomial observations (1/overdispersion) 5.139 5.331
Precision for stock 0.019 0.027
mode
size for the nbinomial observations (1/overdispersion) 5.141
Precision for stock 0.019
Deviance Information Criterion (DIC) ...............: 81090.93
Deviance Information Criterion (DIC, saturated) ....: 155260.91
Effective number of parameters .....................: 63.00
Watanabe-Akaike information criterion (WAIC) ...: 81094.76
Effective number of parameters .................: 65.48
Marginal log-Likelihood: -45533.14
is computed
Posterior summaries for the linear predictor and the fitted values are computed
(Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')
5.4 Comparing
5.4.1 DIC
5.4.2 WAIC
